Abstract

This study extends the imaging, magnification, and aberration properties of the images formed from holograms to the nonparaxial case. This extension yields expressions which can be used to locate accurately the position of an image point in those cases where the paraxial theory fails. The readout-illumination critical angles for image formation are predicted and related to the critical angles of a diffraction grating. The ratio of readout wavelength to recording wavelength is limited, and consequently wavelength-change magnification is limited, for certain types of holograms. The expressions developed are kept in the same form as those developed for the paraxial case so that the reduction to the paraxial case is obvious, and the many prior comments made on aberration reduction are applicable by a simple change of variable.

© 1967 Optical Society of America

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References

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    [Crossref]
  6. I. S. Sokolnikoff, Advanced Calculus (McGraw-Hill Book Co., New York, 1939), p. 303.
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    [Crossref]
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  9. E. L. O’Neill, Introduction to Statistical Optics (Addison–Wesley Pub. Co., Reading, Mass., 1963), p. 47.
  10. M. Born and E. Wolf, Principles of Optics (Pergamon Press, New York, 1964), 2nd ed., pp. 217–218.
  11. E. L. O’Neill, Ref. 9, p. 52.
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    [Crossref]
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    [Crossref]

1966 (3)

1965 (3)

1962 (1)

1949 (1)

D. Gabor, Proc. Roy. Soc. (London) A197, 454 (1949); Proc. Phys. Soc. (London) A64, 449 (1951).

Armstrong, J.

J. Armstrong, IBM J. Res. Develop. 9, 171 (1965).
[Crossref]

Born, M.

M. Born and E. Wolf, Principles of Optics (Pergamon Press, New York, 1964), 2nd ed., pp. 217–218.

DeVelis, J. B.

Gabor, D.

D. Gabor, Proc. Roy. Soc. (London) A197, 454 (1949); Proc. Phys. Soc. (London) A64, 449 (1951).

Haines, K. A.

Leith, E. N.

Meier, R. W.

Neumann, D. B.

O’Neill, E. L.

E. L. O’Neill, Introduction to Statistical Optics (Addison–Wesley Pub. Co., Reading, Mass., 1963), p. 47.

E. L. O’Neill, Ref. 9, p. 52.

Parrent, G. B.

Sokolnikoff, I. S.

I. S. Sokolnikoff, Advanced Calculus (McGraw-Hill Book Co., New York, 1939), p. 303.

Stroke, G. W.

G. W. Stroke, in Optical and Electro-Optical Information Processing, J. T. Tippett, D. A. Berkowitz, L. C. Clapp, C. J. Koester, and A. Vanderburgh, Eds. (MIT Press, Cambridge, Mass., 1965), p. 762.

Thompson, B. J.

Upatnieks, J.

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Pergamon Press, New York, 1964), 2nd ed., pp. 217–218.

IBM J. Res. Develop. (1)

J. Armstrong, IBM J. Res. Develop. 9, 171 (1965).
[Crossref]

J. Opt. Soc. Am. (6)

Proc. Roy. Soc. (London) (1)

D. Gabor, Proc. Roy. Soc. (London) A197, 454 (1949); Proc. Phys. Soc. (London) A64, 449 (1951).

Other (5)

G. W. Stroke, in Optical and Electro-Optical Information Processing, J. T. Tippett, D. A. Berkowitz, L. C. Clapp, C. J. Koester, and A. Vanderburgh, Eds. (MIT Press, Cambridge, Mass., 1965), p. 762.

E. L. O’Neill, Introduction to Statistical Optics (Addison–Wesley Pub. Co., Reading, Mass., 1963), p. 47.

M. Born and E. Wolf, Principles of Optics (Pergamon Press, New York, 1964), 2nd ed., pp. 217–218.

E. L. O’Neill, Ref. 9, p. 52.

I. S. Sokolnikoff, Advanced Calculus (McGraw-Hill Book Co., New York, 1939), p. 303.

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Figures (4)

Fig. 1
Fig. 1

Coordinate location of point Q.

Fig. 2
Fig. 2

Location and orientation of (u,v) plane.

Fig. 3
Fig. 3

Focus angles of hologram 1.

Fig. 4
Fig. 4

Axial-focus distance of hologram 2.

Tables (1)

Tables Icon

Table I Observed real-image focus distances of hologram as a function of αc.

Equations (32)

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E q = ( A q / r q ) exp { i ( 2 π / λ q ) r q }
E 1 = A 1 exp { i ( 2 π / λ c ) r c } exp { i ( 2 π / λ r ) ( r 0 - r r ) m }
E 2 = A 2 exp { i ( 2 π / λ c ) r c } exp { - i ( 2 π / λ r ) ( r 0 - r r ) m }
E I = A I exp { i ( 2 π / λ c ) r I }
r q = [ z q 2 + ( x - x q ) 2 + ( y - y q ) 2 ] 1 2 .
r q = [ x 2 + y 2 - 2 x x q - 2 y y q + R q 2 ] 1 2 ,
r q = R q + ( x 2 + y 2 ) / 2 R q - x x q / R q - y y q / R q - ( 1 / 8 R q 3 ) [ ( x 2 + y 2 ) 2 - 4 ( x 2 + y 2 ) ( x x q + y y q ) + 4 ( x x q + y y q ) 2 ] + ,
1 R I = 1 R c ± ( μ m 2 ) ( 1 R 0 - 1 R r ) ,
x I R I = x c R c ± ( μ m ) ( x 0 R 0 - x r R r )
sin α I = sin α c ± ( μ / m ) ( sin α 0 - sin α r ) ;
y I R I = y c R c ± ( μ m ) ( y 0 R 0 - y r R r )
cos α I sin β I = cos α c sin β c ± ( μ / m ) × ( cos α 0 sin β 0 - cos α r sin β r ) .
( x I / R I ) 2 + ( y I / R I ) 2 < 1
Δ α I / Δ α 0 = ± ( μ / m ) ( cos α 0 / cos α I )
Δ β I / Δ β 0 = ± ( μ / m ) ( cos α 0 cos β 0 / cos α I cos β I ) .
M I α = R I Δ α I / R 0 Δ α 0 = ( cos α 0 / cos α I ) [ m / ( 1 ± m 2 R 0 / μ R c - R 0 / R r ]
M I β = R I cos α I Δ β I / R 0 cos α 0 Δ β 0 = ( cos β 0 / cos β I ) [ m / ( 1 ± m 2 R 0 / μ R c - R 0 / R r ) ]
M long = ± ( 1 / μ ) [ m / ( 1 ± m 2 R 0 / μ R c - R 0 / R r ) ] 2 .
Φ 3 = ( 2 π / λ c ) [ - 1 8 ρ 4 S + 1 2 ρ 3 ( cos θ C x + sin θ C y ) - 1 2 ρ 2 ( cos 2 θ A x + sin 2 θ A y + 2 cos θ sin θ A x y ) ] .
S = 1 R c 3 ± ( μ m 4 ) ( 1 R 0 3 - 1 R r 3 ) - 1 R I 3 .
C x = x c R c 3 ± ( μ m 3 ) ( x 0 R 0 3 - x r R r 3 ) - x I R I 3 ,
C y = y c R c 2             ( μ m 3 ) ( y 0 R 0 3 - y r R r 3 ) - y I R I 3 .
A x = x c 2 R c 3 ± ( μ m 2 ) ( x 0 2 R 0 3 - x r 2 R r 3 ) - x I 2 R I 3 ,
A y = y c 2 R c 2 ± ( μ m 2 ) ( y 0 2 R 0 3 - y r 2 R r 3 ) - y I 2 R I 3 ,
A x y = x c y c R c 3 ± ( μ m 2 ) ( x 0 y 0 R 0 3 - x r y r R r 3 ) - x I y I R I 3 .
| x q y q z q | = | cos α q 0 - sin α q - sin α q sin β q cos β q - cos α q sin β q sin α q cos β q sin β q cos α q cos β q | | u q v q - R q | .
1 / R I = 1 / R c ± ( 1 / R 0 - 1 / R r )
sin α I = sin α c + ( sin α 0 - sin α r ) .
sin α I = sin α c ± 2 sin 17° .
tan α I = tan α c ± 2 tan 17° .
1 / z I = 1 / z c - 1 / z 0 .
sin α c = ± sin 17° .